vygotsky's everyday concepts/scientific concepts dialectics in school ...

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to promote students' argumentation as well as the use of specific symbolic representations. In the "fields of experience" didactical setting, a fairly common classroom routine consists: of individual production of written hypotheses on a given task; of classroom comparison and discussion of selected students' products, orchestrated by the teacher (cf Bartolini Bussi, 1996); of individual written reports about the discussion; and of a classroom summary, usually constructed under the guide of the teacher and finally written down in the students' copybooks. In most cases, classroom summaries represent students’ reached knowledge (with possible ambiguities and hidden mistakes). Final institutionalisation phases (Brousseau, 1986) are attained only in few circumstances. From the researcher's point of view, this style of slowly evolving knowledge offers the opportunity to observe the transformation of students' knowledge in a favourable climate. Argumentation has a strong status in this didactical setting, and most problem situations, particularly in mathematics, are solved through argumentation, avoiding manipulative activities (cf Douek, 2003). Argumentation reflects fairly well each student’s level of mastery of knowledge, his/her level of use of references, and the class' common stable references (cf Douek, 1999). Justification and explanation are essential in the didactical contract. As in all Italian primary schools, the student group remains with the same two teachers for the first five years of primary education. Therefore didactical contracts are very stable. I must add that Italian curricula are rather loosely prescriptive around a core of necessary knowledge. Students deal with the plant cultivation experience field for about 4 months in Grade II. Students had measured wheat plants taken in a field, then had to follow the increase in time of the heights of plants of the classroom pot. They described them and drew them on their copybooks. They solved various problems concerning their growing up, which involved mathematical knowledge; our example concerns one of those problem situations. We will consider data deriving from direct observation, students' texts, and videos of classroom discussions concerning this problem situation as well as previous activities. THE CASE AT STUDY Here is the problem posed to the class: On Friday December 3, the plant in the pot, kept in class and in the light, was 26 cm tall, and the one in the dark was 7 cm tall. Today, Friday December 10, the plant in the pot kept in the light is 28 cm tall and the one in the pot in the dark is 10 cm tall. Which plant has grown the most from December 3 till now? Students had to deal with two kinds of conceptual difficulties. The first difficulty is related to the comparison of transformations (increases) in the additive conceptual field, a complex and new situation for them (cf Vergnaud’s categorisation, in Vergnaud, 1990). The second difficulty is inherent in the students' concept of increase, which was at the level of an everyday concept, although students were able to solve problems as “how much such plant has increased from that date till now”,
and to compare heights of two plants. The closeness between the concepts of height and increase added complexity. However, it will be interesting to see how the evolution of the conceptualisation of increase depended on, and simultaneously favoured, the mastery of the additive field and its extension to include the comparison of transformations. Phase 1: Individual oral production with teacher’s mediation 13 students (out of 20) engaged correctly into the problem solving, through calculations and comparison of results. But they were confused when the teacher asked them to explain their problem solving. Only 6 of them arrived easily at a stable correct conclusion. Some students answered “the plant that increased the most is the one which is the tallest”. They deduced increase from height, thus contradicting their own numerical result. Others said “this plant increased the most, so it is the tallest”. They deduced height from increase, disregarding the visible state of the plants. Performing the right calculations was not enough to grasp the situation as a whole. To solve problem numerically, most students interpreted increase as a supplementary length in the plant. But the dynamical character of increase was not grasped with its dependence on time and the implicit fact that it may be slow or quick between two given moments: they were not able to use such an argument to interpret the situation. Their everyday conceptualisation of increase was not sufficient. But through argumentation guided (or stimulated, depending on cases) by the teacher, this concept evolved and most students became able, at least momentarily, to handle the problem situation through verbal activity, paying attention to the various elements of the situation, and were able to interpret them according to a hierarchy that organised calculations. Many students used an expression that became meaningful and stabilised: “pezzo cresciuto” ("the bit that grew"). The other 7 students had difficulties to master and interpret data, and did not succeed in producing a solution. The teacher encouraged them (as well as 2 students out of the group of 13 who had produced a correct solution) to draw the two plants, mark the drawing with signs corresponding to the different heights at different dates, and follow the increases on the drawing. Hence for all the students the situation gained a spatial character, either through the drawing or through the image of “the bit that grew” (which is not correct scientifically, since growth is an evolution of the whole plant). This spatial frame helped most students to “see” and grasp the dynamical aspect of the plant growing as well as to fix for each plant the elements that need to be determined and compared. Given students' mastery of the additive field, I conjecture that those spatial representations allowed the representation of numbers on the number line to be transferred (gradually, depending on students) to this new additive situation. This favoured the mathematisation of this new situation while putting its dynamical character into new evidence. As a whole, student/teacher interactions developed argumentation concerning three types of references: reality of plants cultivation, their “stories” with their dynamical character and time dependence; schematisation through drawing and/or its equivalent
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Page 5 and 6: arguments to interpret and compare
Page 7 and 8: they can be identified through the

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